# Proof of Main Theorem

Lemma 2   : Let be a square-free integer, be the greatest integer in , as defined in (4) and . Then if and only if Proof: For a proof, see [??].

Theorem 3 (B = 2C)   : Let be a square-free integer such that the continued fraction representation of has a center b, be the greatest integer in , , and be defined as in (4). Then if and only if .

Proof: Suppose that . Then so by Lemma 2,  Carrying out this fractional linear transformation yields Substituting in for and making the substitution stated in (8) gives the equation But by the construction of , det = . Thus making the substitution into the above equation, we get the quadratic equation (in B) Solving for yields (9)

But by the choice of and the construction of and , we have that and thus the second possibility must be ruled out because it would yield a negative value for . Hence as required. Conversly, assume that . Then it is true that But once again, so if we make the substitution , it is true that Then making the substitution and the one stated in (8) yields (after rearranging terms) It follows that and thus by Lemma 2, and . For the remainder of the report we consider those continued fractions such that . Then we can make a change of variables in the matrix in (4) and let (10)

scanez 2000-12-04